Efficient time discretization scheme for nonlinear space fractional reaction–diffusion equations

ABSTRACT

We present a novel Exponential Time Differencing (ETD) scheme for nonlinear Riesz space fractional reaction–diffusion equations. This scheme is based on using a real distinct poles (RDP) discretization for the underlying matrix exponentials. Due to these RDP, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second-order convergent; and proven to be robust for problems involving non-smooth/mismatched initial and boundary conditions and steep solution gradients. We examine the stability of the scheme through its amplification factor and plot the boundaries of the stability regions comparative to other second-order ETD schemes. This numerical scheme combined with fractional central differencing is used for simulating some nonlinear space fractional problems. We demonstrate the superiority of our method over competing second-order ETD schemes, BDF2 scheme, and IMEX schemes. Our experiments show that the proposed scheme is computationally more efficient (in terms of cpu time). Furthermore, we investigate the trade-off between using fractional central differencing and matrix transfer technique in discretization of Riesz fractional derivatives.

KEYWORDS:

• Riesz fractional derivative
• exponential time differencing
• real distinct pole
• matrix transfer
• reaction diffusion equations

2010 AMS SUBJECT CLASSIFICATIONS:

• 35K57
• 35K51
• 65M06
• 65M20
• 45G15

Disclosure statement

No potential conflict of interest was reported by the authors.